44 



ON THE EQUATION OF CURVES 



good unless the point A he on a straight line drawn through O 

 parallel to one of the asymptotes, in which case the points Pj 

 and Q, coincide. 



'3 



VII. 



Let us now consider the case in which the curve (1) is a 

 parabola. Let P, P^ (Fig. 6) be an ordinate to the diameter 

 AP3, then since AP^^AP^ (11), equation (15) gives 

 AP 2 OQ,.OQ, 



hence, if AV ^=:^x, APj=y, we shall have 



y'^:=p'x (16) 



for the equation of the parabola referred to any diameter AQ, 

 and the line drawn through Q3 parallel to its ordinates, as 

 axes of co-ordinates. 



(a.) The quantity p' evidently remains invariable for the 

 same diameter, and is called the parameter of that diameter. 

 To find a general expression for p', we have (i) 



OQ, .OQ, ^ F(m^+g^^cosy+I ) 

 ' * Am^+^Bw+C ' 



no _ — ^V {m^^+2m^ cosy+1) 



— F/ (B^— 2BCcosy-|-C^) . C 



= 2(BE-CD) ^ ^^"^^ ^--B • 



2{ClD—BE){m^ + 2mcosy-{-\) 

 *'• ■^"~(A7n^+2Bm+C)-/(B^— 2BCcosy+C2)'*"^ ^* 

 m being the direction index of the ordinates to the diameter 

 AP3. 



(j3.) The vertex P3 of the diameter AP3 may be determined 

 from the simultaneous equations (1) and (10.) Combining 

 these with the equation B^ = AC, we obtain, by eliminating 

 C and y, and then eliminating A and x, 



AF— D2 , BD— AE 



2(BD— AE) 2[Am+B)^ 

 CF— E* , (BE— CD) w^ 



2 (BE— CD) 2(Bm+C) 



.(18), 



